1. Chapter 5
Mathematics of Finance

2. INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
Applications and More Algebra
Functions and Graphs
Lines, Parabolas, and Systems
Exponential and Logarithmic Functions
Mathematics of Finance
Matrix Algebra
Linear Programming
Introduction to Probability and Statistics

3. INTRODUCTORY MATHEMATICAL ANALYSIS
Additional Topics in Probability
Limits and Continuity
Differentiation
Additional Differentiation Topics
Curve Sketching
Integration
Methods and Applications of Integration
Continuous Random Variables
Multivariable Calculus

4. Chapter 5: Mathematics of Finance
Chapter Objectives
To solve interest problems which require logarithms.
To solve problems involving the time value of money.
To solve problems with interest is compounded continuously.
To introduce the notions of ordinary annuities and annuities due.
To learn how to amortize a loan and set up an amortization schedule.

5. Chapter Outline Compound Interest 5.1) Present Value
Chapter 5: Mathematics of Finance
Chapter Outline
Compound Interest
Present Value
Interest Compounded Continuously
Annuities
Amortization of Loans
5.1)
5.2)
5.3)
5.4)
5.5)

6. Chapter 5: Mathematics of Finance
5.1 Compound Interest
Example 1 – Compound Interest
Compound amount S at the end of n interest periods at the periodic rate of r is as
Suppose that $500 amounted to $ in a savings account after three years. If interest was compounded semiannually, find the nominal rate of interest, compounded semiannually, that was earned by the money.

7. There are 2 × 3 = 6 interest periods.
Chapter 5: Mathematics of Finance
5.1 Compound Interest
Example 1 – Compound Interest
Solution:
There are 2 × 3 = 6 interest periods.
The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.

8. The periodic rate is r = 0.06/4 = 0.015.
Chapter 5: Mathematics of Finance
5.1 Compound Interest
Example 3 – Compound Interest
How long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly?
Solution:
The periodic rate is r = 0.06/4 =
It will take

9. Example 5 – Effective Rate Effective Rate
Chapter 5: Mathematics of Finance
5.1 Compound Interest
Example 5 – Effective Rate
Effective Rate
The effective rate re for a year is given by
To what amount will $12,000 accumulate in 15 years if it is invested at an effective rate of 5%?
Solution:

10. Respective effective rates of interest are
Chapter 5: Mathematics of Finance
5.1 Compound Interest
Example 7 – Comparing Interest Rates
If an investor has a choice of investing money at 6% compounded daily or % compounded quarterly, which is the better choice?
Solution:
Respective effective rates of interest are
The 2nd choice gives a higher effective rate.

11. Chapter 5: Mathematics of Finance
5.2 Present Value
Example 1 – Present Value
P that must be invested at r for n interest periods so that the present value, S is given by
Find the present value of $1000 due after three years if the interest rate is 9% compounded monthly.
Solution:
For interest rate,
Principle value is

12. a final payment at the end of five years.
Chapter 5: Mathematics of Finance
5.2 Present Value
Example 3 – Equation of Value
A debt of $3000 due six years from now is instead to be paid off by three payments:
$500 now,
$1500 in three years, and
a final payment at the end of five years.
What would this payment be if an interest rate of 6% compounded annually is assumed?

13. The equation of value is
Chapter 5: Mathematics of Finance
5.2 Present Value
Solution:
The equation of value is

14. Example 5 – Net Present Value
Chapter 5: Mathematics of Finance
5.2 Present Value
Example 5 – Net Present Value
Net Present Value
You can invest $20,000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table.
Assume an interest rate of 7% compounded annually and find the net present value of the cash flows.
Year
Cash Flow 2
$10,000 3
8000 5
6000

15. Solution: Chapter 5: Mathematics of Finance 5.2 Present Value
Example 5 – Net Present Value
Solution:

16. 5.3 Interest Compounded Continuously
Chapter 5: Mathematics of Finance
5.3 Interest Compounded Continuously
Example 1 – Compound Amount
Compound Amount under Continuous Interest
The compound amount S is defined as
If $100 is invested at an annual rate of 5% compounded continuously, find the compound amount at the end of
a. 1 year.
b. 5 years.

17. Effective Rate under Continuous Interest
Chapter 5: Mathematics of Finance
5.3 Interest Compounded Continuously
Effective Rate under Continuous Interest
Effective rate with annual r compounded continuously is
Present Value under Continuous Interest
Present value P at the end of t years at an annual r compounded continuously is .

18. We want the present value of $25,000 due in 20 years.
Chapter 5: Mathematics of Finance
5.3 Interest Compounded Continuously
Example 3 – Trust Fund
A trust fund is being set up by a single payment so that at the end of 20 years there will be $25,000 in the fund. If interest compounded continuously at an annual rate of 7%, how much money should be paid into the fund initially?
Solution:
We want the present value of $25,000 due in 20 years.

19. 5.4 Annuities Sequences and Geometric Series
Chapter 5: Mathematics of Finance
5.4 Annuities
Example 1 – Geometric Sequences
Sequences and Geometric Series
A geometric sequence with first term a and common ratio r is defined as
a. The geometric sequence with a = 3, common ratio 1/2 , and n = 5 is

20. b. Geometric sequence with a = 1, r = 0.1, and n = 4.
Chapter 5: Mathematics of Finance
5.4 Annuities
Example 1 – Geometric Sequences
b. Geometric sequence with a = 1, r = 0.1, and
n = 4.
c. Geometric sequence with a = Pe−kI , r = e−kI ,
n = d.
Sum of Geometric Series
The sum of a geometric series of n terms, with first term a, is given by

21. Find the sum of the geometric series:
Chapter 5: Mathematics of Finance
5.4 Annuities
Example 3 – Sum of Geometric Series
Find the sum of the geometric series:
Solution: For a = 1, r = 1/2, and n = 7
Present Value of an Annuity
The present value of an annuity (A) is the sum of the present values of all the payments.

22. Chapter 5: Mathematics of Finance
5.4 Annuities
Example 5 – Present Value of Annuity
Find the present value of an annuity of $100 per month for years at an interest rate of 6% compounded monthly.
Solution:
For R = 100, r = 0.06/12 = 0.005, n = ( )(12) = 42
From Appendix B, .
Hence,

23. Chapter 5: Mathematics of Finance
5.4 Annuities
Example 7 – Periodic Payment of Annuity
If $10,000 is used to purchase an annuity consisting of equal payments at the end of each year for the next four years and the interest rate is 6% compounded annually, find the amount of each payment.
Solution:
For A= $10,000, n = 4, r = 0.06,

24. The amount S of ordinary annuity of R for n periods at r per period is
Chapter 5: Mathematics of Finance
5.4 Annuities
Example 9 – Amount of Annuity
Amount of an Annuity
The amount S of ordinary annuity of R for n periods at r per period is
Find S consisting of payments of $50 at the end of every 3 months for 3 years at 6% compounded quarterly. Also, find the compound interest.
Solution: For R=50, n=4(3)=12, r=0.06/4=0.015,

25. Chapter 5: Mathematics of Finance
5.4 Annuities
Example 11 – Sinking Fund
A sinking fund is a fund into which periodic payments are made in order to satisfy a future obligation. A machine costing $7000 is replaced at the end of 8 years, at which time it will have a salvage value of $700. A sinking fund is set up. The amount in the fund at the end of 8 years is to be the difference between the replacement cost and the salvage value. If equal payments are placed in the fund at the end of each quarter and the fund earns 8% compounded quarterly, what should each payment be?

26. Amount needed after 8 years = 7000 − 700 = $6300.
Chapter 5: Mathematics of Finance
5.4 Annuities
Example 11 – Sinking Fund
Solution:
Amount needed after 8 years = 7000 − 700 = $6300.
For n = 4(8) = 32, r = 0.08/4 = 0.02, and S = 6300, the periodic payment R of an annuity is

27. 5.5 Amortization of Loans Amortization Formulas
Chapter 5: Mathematics of Finance
5.5 Amortization of Loans
Amortization Formulas

28. total interest charges, and principal remaining after five years.
Chapter 5: Mathematics of Finance
5.5 Amortization of Loans
Example 1 – Amortizing a Loan
A person amortizes a loan of $170,000 by obtaining a 20-year mortgage at 7.5% compounded monthly. Find
monthly payment,
total interest charges, and
principal remaining after five years.

29. Solution: a. Monthly payment: b. Total interest charge:
Chapter 5: Mathematics of Finance
5.5 Amortization of Loans
Example 1 – Amortizing a Loan
Solution:
a. Monthly payment:
b. Total interest charge:
c. Principal value: